Performance Evaluation of Variational Quantum Eigensolver and Quantum Dynamics Algorithms on the Advection-Diffusion Equation

TL;DR

The paper benchmarks near-term quantum PDE solvers by comparing a VQE-based linear-system approach against quantum dynamics methods (Trotterization, VarQTE, AVQDS) for a 1D advection-diffusion equation, using a noiseless statevector baseline to isolate algorithmic performance. It demonstrates that, for $N=4$ qubits, the VQE solver can reach final-time fidelities as good as $\mathcal{O}(10^{-9})$ with moderate circuit depth, outperforming hardware runs that exhibit fidelities around $\mathcal{O}(10^{-1})$. The work provides a rigorous comparison of resource demands, showing VQE requires far fewer entangling gates than Trotter-based methods and offers a clean algorithmic baseline for PDE solvers on NISQ hardware. It also outlines clear paths to bridge toward real hardware performance, including noise modeling, error mitigation, and extension to higher dimensions and nonlinear PDEs relevant to engineering and finance.

Abstract

We investigate the potential of near-term quantum algorithms for solving partial differential equations (PDEs), focusing on a linear one-dimensional advection-diffusion equation as a test case. This study benchmarks a ground-state algorithm, Variational Quantum Eigensolver (VQE), against three leading quantum dynamics algorithms, Trotterization, Variational Quantum Imaginary Time Evolution (VarQTE), and Adaptive Variational Quantum Dynamics Simulation (AVQDS), applied to the same PDE on small quantum hardware. While Trotterization is fully quantum, VarQTE and AVQDS are variational algorithms that reduce circuit depth for noisy intermediate-scale quantum (NISQ) devices. However, hardware results from these dynamics methods show sizable errors due to noise and limited shot statistics. To establish a noise-free performance baseline, we implement the VQE-based solver on a noiseless statevector simulator. Our results show VQE can reach final-time infidelities as low as ${O}(10^{-9})$ with $N=4$ qubits and moderate circuit depths, outperforming hardware-deployed dynamics methods that show infidelities $\gtrsim 10^{-1}$. By comparing noiseless VQE to shot-based and hardware-run algorithms, we assess their accuracy and resource demands, providing a baseline for future quantum PDE solvers. We conclude with a discussion of limitations and potential extensions to higher-dimensional, nonlinear PDEs relevant to engineering and finance.

Figures (2)

• Figure 1: (a) Scalar field $C(x)$ reconstructed from VQE outputs at various time points ($t = 0$, $0.25$, $0.75$, $1.0$), alongside the corresponding DNS result at $t = 1.0$ (dashed black line). (b) Infidelity $1 - f$ between the VQE and DNS solutions vs. time for different circuit layers (1 to 10, from top to bottom). At 5 layers, $1-f$ drops below $10^{-5}$. Beyond 5 layers, it can reach $10^{-9}$. (c) Final-time infidelity ($t=1$) vs. number of layers.
• Figure 2: Quantum circuit of the real-amplitudes VQE ansatz for $N=4$ qubits with one layer, alongside its transpilation for a linear qubit topology. Each layer has single-qubit $R_y(\theta)$ gates followed by CNOTs between adjacent qubits.